Narayana sequence and the Brocard–Ramanujan equation

Mustafa Ismail, Salah Eddine Rihane and M. Anwar
Notes on Number Theory and Discrete Mathematics
Print ISSN 1310–5132, Online ISSN 2367–8275
Volume 29, 2023, Number 3, Pages 462–473
DOI: 10.7546/nntdm.2023.29.3.462-473
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Authors and affiliations

Mustafa Ismail
Department of Mathematics, Faculty of Science,
University of Ain Shams, Egypt

Salah Eddine Rihane
Department of Mathematics, Institute of Science and Technology,
University Center of Mila, Algeria

M. Anwar
Department of Mathematics, Faculty of Science,
University of Ain Shams, Egypt

Abstract

Let \left\lbrace a_{n}\right\rbrace_{n\geq 0} be the Narayana sequence defined by the recurrence a_{n}=a_{n-1}+a_{n-3} for all n\geq 3 with intital values a_{0}=0 and a_{1}=a_{2}=1. In this paper, we fully characterize the 3-adic valuation of a_{n}+1 and a_{n}-1 and then we find all positive integer solutions (u,m) to the Brocard–Ramanujan equation m!+1=u^2 where u is a Narayana number.

Keywords

  • Narayana sequence
  • Factorials
  • p-adic valuation

2020 Mathematics Subject Classification

  • 11B39
  • 11D72

References

  1. Allouche, J. P., & Johnson, T. (1996). Narayana’s cows and delayed morphisms. Journées d’Informatique Musicale, ˆıle de Tatihou, France. Available online at: https://hal.science/hal-02986050
  2. Berndt, B. C., & Galway, W. F. (2000). On the Brocard–Ramanujan Diophantine equation n! + 1 = m^2. The Ramanujan Journal, 4(2), 41–42.
  3. Bilgici, G. (2016). The generalized order-k Narayana’s cows numbers. Mathematica Slovaca, 66(4), 795–802.
  4. Brocard, H. (1876). Question 166, Nouv. Corresp. Math., 2, 287.
  5. Facó, V., & Marques, D. (2016). Tribonacci numbers and the Brocard–Ramanujan equation. Journal of Integer Sequences, 19(4), 4–16.
  6. Grossman, G., & Luca, F. (2002). Sums of factorials in binary recurrence sequences. Journal of Number Theory, 93(2), 87–107.
  7. Luca, F. (1999). Products of factorials in binary recurrence sequences. The Rocky Mountain Journal of Mathematics, 29(4), 1387–1411.
  8. Luca, F., & Stanica, P. (2006). {F_{1}F_{2}F_{3}F_{4}F_{5}F_{6}F_{8}F_{10}F_{12}}= 11!. Portugaliae Mathematica, 63(3), 251–260.
  9. Marques, D. (2012). Fibonacci numbers at most one away from the product of factorials. Notes on Number Theory and Discrete Mathematics, 18(3), 13–19.
  10. Marques, D. (2012). The order of appearance of product of consecutive Fibonacci numbers. Fibonacci Quarterly, 50(2), 132–139.
  11. Overholt, M. (1993). The Diophantine equation n!+ 1= m^2. Bulletin of the London Mathematical Society, 25(2), 104.
  12. Ramanujan, S. (1962). Ramanujan’s Collected Works. Chelsea, New York.
  13. Zhou, C. (1999). Constructing identities involving Kth-order F-L numbers by using the characteristic polynomial. Applications of Fibonacci Numbers, 369–379.

Manuscript history

  • Received: 20 July 2022
  • Revised: 25 April 2023
  • Accepted: 27 June 2023
  • Online First: 6 July 2023

Copyright information

Ⓒ 2023 by the Authors.
This is an Open Access paper distributed under the terms and conditions of the Creative Commons Attribution 4.0 International License (CC BY 4.0).

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Cite this paper

Ismail, M., Rihane, S. E., Anwar, M. (2023). Narayana sequence and the Brocard–Ramanujan equation. Notes on Number Theory and Discrete Mathematics, 29(3), 462-473, DOI: 10.7546/nntdm.2023.29.3.462-473.

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